p-Harmonic measure is not additive on null sets

Llorente, José G.; Manfredi, Juan J.; Wu, Jun

doi:10.2422/2036-2145.2005.2.06

Cited by 11 publications

(14 citation statements)

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“…At the same time, although not directly applicable here, the main result (Theorem 1.1) in Llorente-Manfredi-Wu [46] indicates that in general Q W is not likely to be a vector lattice. Their result shows that for the upper half plane (which is unbounded and thus not included here) whenever 1 < p < ∞ and p = 2, there are finitely many sets E 1 , E 2 , ... , E n such that R = n j=1 E j while P χ Ej ≡ 0 for all j and P χ R ≡ 1.…”

Section: It Will Be Convenient To Define Lmentioning

confidence: 84%

The Dirichlet problem for $p$-harmonic functions with respect to arbitrary compactifications

Björn

Sjödin

2018

Rev. Mat. Iberoam.

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We study the Dirichlet problem for p-harmonic functions on metric spaces with respect to arbitrary compactifications. A particular focus is on the Perron method, and as a new approach to the invariance problem we introduce Sobolev-Perron solutions. We obtain various resolutivity and invariance results, and also show that most functions that have earlier been proved to be resolutive are in fact Sobolev-resolutive. We also introduce (Sobolev)-Wiener solutions and harmonizability in this nonlinear context, and study their connections to (Sobolev)-Perron solutions, partly using Q-compactifications.

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Section: It Will Be Convenient To Define Lmentioning

confidence: 84%

The Dirichlet problem for $p$-harmonic functions with respect to arbitrary compactifications

Björn

Sjödin

2018

Rev. Mat. Iberoam.

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“…When studying boundary behavior of p-harmonic type functions, various versions of generalizations of harmonic measures have been introduced and studied for p = 2, see e.g. Llorente-Manfredi-Wu [49]. In the case of constant p (p = 2) Bennewitz and Lewis employed the doubling property of a p-harmonic measure, first proved in Eremenko-Lewis [26], to obtain a Boundary Harnack inequality for p-harmonic functions in the plane, see Bennewitz-Lewis [17].…”

Section: P(•)-harmonic Measurementioning

confidence: 99%

The boundary Harnack inequality for variable exponent $$p$$ p -Laplacian, Carleson estimates, barrier functions and $${p(\cdot )}$$ p ( · ) -harmonic measures

Adamowicz

Lundström

2015

Annali di Matematica

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We investigate various boundary decay estimates for p(·)-harmonic functions. For domains in R n , n ≥ 2 satisfying the ball condition (C 1,1 -domains), we show the boundary Harnack inequality for p(·)-harmonic functions under the assumption that the variable exponent p is a bounded Lipschitz function. The proof involves barrier functions and chaining arguments. Moreover, we prove a Carleson-type estimate for p(·)-harmonic functions in NTA domains in R n and provide lower and upper growth estimates and a doubling property for a p(·)-harmonic measure.

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“…When F is the characteristic function 1 A for some A ⊂ ∂Ω, the value H F (x) is called the p-harmonic measure of A at x and written ω p (A, x, Ω). It is well known (see, e.g., [6]) that if ω p (A, x 0 , Ω) = 0 for some x 0 in a connected domain Ω, then ω p (A, x, Ω) = 0 for all points x ∈ Ω. Theorem 4.2 is interesting in light of the fact that p-harmonic measure is non-additive even on null sets [11]. In fact, [11] exhibits a disjoint finite collection {A i } of resolutive sets with p-harmonic measure zero whose union is all of ∂Ω.…”

Section: Shrinking Step Sizesmentioning

confidence: 99%

Tug-of-war with noise: A game-theoretic view of the p-Laplacian

2008

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Fix a bounded domain Ω ⊂ R d , a continuous function F : ∂Ω → R, and constants ǫ > 0 and 1 < p, q < ∞ with p −1 + q −1 = 1. For each x ∈ Ω, let u ǫ (x) be the value for player I of the following two-player, zero-sum game. The initial game position is x. At each stage, a fair coin is tossed and the player who wins the toss chooses a vector v ∈ B(0, ǫ) to add to the game position, after which a random "noise vector" with mean zero and variance q p |v| 2 in each orthogonal direction is also added. The game ends when the game position reaches some y ∈ ∂Ω, and player I's payoff is F (y).We show that (for sufficiently regular Ω) as ǫ tends to zero the functions u ǫ converge uniformly to the unique p-harmonic extension of F . Using a modified game (in which ǫ gets smaller as the game position approaches ∂Ω), we prove similar statements for general bounded domains Ω and resolutive functions F .These games and their variants interpolate between the tug of war games studied by Peres, Schramm, Sheffield, and Wilson (p = ∞) and the motion-by-curvature games introduced by Spencer and studied by Kohn and Serfaty (p = 1). They generalize the relationship between Brownian motion and the ordinary Laplacian and yield new results about p-capacity and p-harmonic measure.

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p-Harmonic measure is not additive on null sets

Abstract: When 1 < p < ∞ and p = 2 the p-harmonic measure on the boundary of the half plane R 2

Cited by 11 publications

References 1 publication

The Dirichlet problem for $p$-harmonic functions with respect to arbitrary compactifications

The Dirichlet problem for $p$-harmonic functions with respect to arbitrary compactifications

The boundary Harnack inequality for variable exponent $$p$$ p -Laplacian, Carleson estimates, barrier functions and $${p(\cdot )}$$ p ( · ) -harmonic measures

Tug-of-war with noise: A game-theoretic view of the p-Laplacian

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